Adding more cuts not creating more round object when using curve modifier on specific object
Here is what I found-all that that unnecessary loopcuts were causing the issue. It is probably because you used mirror modifiers to make the barrel. Mirror modifier just make a duplicate of the mesh and mirrors it. It does not merge the vertices to the original mesh.
In your case the easiest way to solve it is by selecting all those edges (which you have beveled) except any one among them by  alt+shift + RMB and then delete them by pressing X.Dont forget to switch off all the modidifiers except curve modifier before selecting


(Select the edge and press Gtwice tp slide the edge)
Which will get u this:

To solve it in the proper way, you can apply the mirror modifier and box select the vertices and merge them with  altM &gt; merge at center .(You wont even see the vertices cuz they are too close to each other).OR simply use the merge option in mirror modifier.